Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 1 - Precalculus Review - 1.3 The Basic Classes of Functions - Exercises - Page 23: 31


$f \circ g (x) = cos(x^3 +x^2)$ $g \circ f (x) = cos^3(x) + cos^2(x)$ Domains: $ \mathbb{R}$

Work Step by Step

$f \circ g$ is defined by $f \circ g (x) = f(g(x)) = cos(x^3 +x^2)$ and $g \circ f$ is defined by $g \circ f (x) = g(f(x)) = cos^3(x) + cos^2(x)$. The domain of $f$ is $\mathbb{R}$ and the domain of $g$ is $\mathbb{R}$. Hence, we have: Domain of $f \circ g$ is given by $D_{f \circ g} = \{x \in D_g : \, g(x) \in D_f\} = \{x \in \mathbb{R} : \, x^3+x^2 \in \mathbb{R}\}$ and then the domain of $f \circ g$ is equal to $\mathbb{R}$. Domain of $g \circ f$ is given by $D_{g \circ f} = \{x \in D_f : \, f(x) \in D_g\} = \{x \in \mathbb{R} : \, cos(x) \in \mathbb{R}\}$ and then the domain of $g \circ f$ is equal to $\mathbb{R}$.
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